Proof of Cauchy's determinant identity

I am reading Matrix Analysis 2nd. Page 66 gives the proof of Caucuh's determinant identity as follow:

where $(1.3.23)$ is as follow:

I am very confused about these questions:

1. How does second identity make sense? What does $\lambda_i$ stand for?
2. The $\lambda_i$ should be mulitplied into $I$ in the third identity. But it did not happen. Why?

Could anyone help me out? Thanks in advance! I am very very confused...

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